Timeline for Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?
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| Jul 10, 2023 at 15:52 | comment | added | S Argyros | I have the following question related to the original one. Suppose that $X$ is a non strictly convexifiable Banach space. Is it true that every dense subspace is also non strictly convexifiable? In particular what is the answer for $l^{\infty} ( \Gamma)$ with $ \Gamma $ an uncountable set. | |
| Jul 9, 2023 at 13:14 | comment | added | S Argyros | The result is due to H.P.Rosenthal. On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13-36. MR 42 #5015. The main lemma is simplified by J. Kupka Proc. AMS Vol. 45 70 - 73 (1974) | |
| Jul 9, 2023 at 12:22 | comment | added | Daron | Oh, you're right, it's not that easy. Does fact 2 have a name? Is it hard to prove? | |
| Jul 9, 2023 at 10:14 | history | edited | S Argyros | CC BY-SA 4.0 |
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| Jul 9, 2023 at 10:00 | comment | added | S Argyros | If you mean my remark then any continuous norm on a Banach space either is equivalent to the original one or the space with the new norm is not complete. | |
| Jul 9, 2023 at 9:51 | comment | added | Daron | I don't fully understand your answer. Are you just using the open mapping theorem for the mapping going from the original space, to the same space with the new norm, to see the two norms are equivalent? | |
| Jul 9, 2023 at 9:44 | history | edited | S Argyros | CC BY-SA 4.0 |
added 246 characters in body
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| Jul 9, 2023 at 9:29 | history | answered | S Argyros | CC BY-SA 4.0 |